Generation of tooth profile for roots rotor based on virtual linkage associated with 

Assur groupCai, Y, Yao, L and Wei, G

http://dx.doi.org/10.1177/1687814016683352

Title Generation of tooth profile for roots rotor based on virtual linkage associated with Assur group

Authors Cai, Y, Yao, L and Wei, G

Type Article

URL This version is available at: http://usir.salford.ac.uk/id/eprint/41138/

Published Date 2016

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Research Article

Advances in Mechanical Engineering2016, Vol. 8(12) 1–8� The Author(s) 2016DOI: 10.1177/1687814016683352aime.sagepub.com

Generation of tooth profile for Rootsrotor based on virtual linkageassociated with Assur group

Yingjie Cai1, Ligang Yao1 and Guowu Wei2

AbstractThis article, for the first time, presents the generation of Roots rotor tooth profiles based on an Assur-group-associatedvirtual linkage method. Taking the original Roots rotor as an example, structure and geometry of the Roots rotor areintroduced, and based on the principle of inversion, an equivalent virtual linkage is identified for generating dedendumtooth profile of the rotor. Using linkage decomposition associated with elemental Assur groups, algorithm for computingthe tooth curve is constructed leading to the explicit expression of rotor profile and the corresponding numerical simu-lation, verifying the validity of the proposed approach. For demonstration purpose, the virtual linkage method is thenextended to the generation of tooth profiles for the variants of Roots rotors with arc-cycloidal curves and arc-involutecurves. Integrated with computer-aided design, computer-aided engineering and computer-aided manufacturing softwareplatforms, as well as the three-dimensional printing technology, this article provides an efficient and intuitive approachfor Roots rotor system design, analysis and development.

KeywordsRoots rotor, Roots blower, tooth profile generation, virtual linkage, Assur group

Date received: 4 March 2016; accepted: 16 November 2016

Academic Editor: ZW Zhong

Introduction

Since the Roots brothers patented the Roots blower in1860, this kind of rotor and its variants have beenwidely used as air dischargers for diesel engines, pumpsand other equipments due to its capability of providinghigher and greater air flow compared to a conventionalpump. In order to bring greater air flow for a Rootsblower, in the past one and a half centuries, differentcross-sectional designs have been proposed,1–6 analysedand developed. Ritchie and Patterson7 investigated thegeometry of involute rotors aiming for minimizing leak-age. Holmes8 and Litvin9 established the mathematicalmodels for the cross sections of the Roots blower rotorswith involute, cycloidal and arc tooth profiles. Litvinand Feng10 proposed computer-aided design (CAD)approach for generating the profiles. Mimmi andPennacchi11,12 investigated different types of positive

displacement rotary pumps and blowers. Liu et al.13

proposed a sealing index and presented a method forthe design of lobe pump with high-sealing property.Furthermore, Tong and Yang3 proposed the pumpingratio, and Yang and Tong14 formulated the flow ratefunctions, leading to the design and profile synthesis ofvarious lobe pumps.15 Vecchiato et al.16 developed therotor cross section as a combination of arcs and a

1School of Mechanical Engineering and Automation, Fuzhou University,

Fuzhou, P.R. China2School of Computing, Science & Engineering, University of Salford,

Salford, UK

Corresponding author:

Ligang Yao, School of Mechanical Engineering and Automation, Fuzhou

University, No. 2, Xueyuan Road, University Town, Fuzhou 350116, P.R.

China.

Email: ylgyao@fzu.edu.cn

Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License

(http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without

further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/

open-access-at-sage).

conjugated epicycloidal curve. Yao et al.17,18 investi-gated the geometry of Roots rotor of arc-cycloidal pro-file and applied the results to the cutting and milling ofthree-lobe helix rotors. Using the concept of variabletrochoid ratio, Hsieh and Hwang4,19,20 developedRoots rotors of high volumetric efficiency and high-sealing property. More recently, Kang and Vu5 pro-posed the epicycloidal-circular-epicycloidal (CEC)rotor profile, and Hsieh6 applied the elliptical roulettecurve to the design of rotary lobe dumps.

Profiles of the cross sections of rotors play crucialrole in the Roots blowers. However, according to theliterature, most of the derivations for Roots rotor toothprofiles are based on gear geometry and theory whichinvolves establishing body-attached reference systems,equation of meshing and coordinate transformationmatrices. For engineers with limited knowledge in geargeometry, such derivations, as those in the treatises ofHolmes8 and Litvin,9 seem complex to them, but gener-ating displacement curve of a linkage through pro-gramming associate with pre-provided Assur-groupsubroutines appears to be easier for them. They caneffectively master and apply the latter approach forRoots rotor profile generating, manufacturing and test-ing. Therefore, this article proposes an Assur-group-associated virtual linkage method for generating thetooth profiles of Roots rotors of various types. The vir-tual linkage method has been used in multi-graspmanipulation,21 haptic manipulation22 and parallelmechanism synthesis,23 but no report has been foundfor its application in generating rotor tooth profiles.

Kempe24 pointed out that planar curves can be gen-erated by linkages; theoretically, the planar curves inthe cross sections of tooth profiles of Roots blowerscan be produced by linkages. Hence, this article, for thefirst time, presents the application of virtual linkagemethod to the generation of Roots rotor tooth profiles.The principle of inversion, as defined in Ye et al.,25 isused in this investigation, leading to the discovery ofequivalent virtual linkages, and programming algo-rithms associated with Assur-group subroutines areformulated and developed, verifying and demonstratingthe validity of the proposed method.

Structure and geometry of a Roots rotor

Figure 1 shows the structure a Roots blower developedin this article. It consists of two rotors, that is, rotor 1and rotor 2. The ratio of the angular velocities betweenthe two rotors is 1; in the practical applications, thistransmission ratio is secured by a pair of equal gearsthat are mounted on the shafts of the two rotors (seeFigure 1). Commonly, the rotors have two or threelobes, and the cross-sectional profiles of the two rotorsare identical as illustrated in Figure 2. The general

profile of a rotor tooth is formed by two types ofcurves, that is, the addendum profile curve, for example,curve K0K1 indicated in Figure 2 which is a circular arcof radius r centred at point C, and the dedendum pro-file curve, for example, curve K 00K 01 which is equidistantto the shortened epicycloid that is commonly generatedthrough equation of meshing.9,18,20 The addendumangle of curve K0K1 of a rotor with two or three lobesequals 90� and 60�, respectively, such that the value ofangle g (see Figure 2) is 45� and 30�, respectively, fortwo- and three-lobe rotors. In this article, we concernonly rotors with three lobes, while the method presentedherein be extended to rotors with two lobes.

From the above description, it can be found that thekey issue to produce the full profile of a rotor lies in thecreation of the dedendum curve K 00K 01. According tothe literature, this curve was, in most of the cases,obtained through the traditional approach based ongear geometry, in which the generation of curve K 00K 01involves establishing coordinate systems, transforma-tion matrices and equation of meshing. In order to sim-plify the derivation and provide an efficient andintuitive way for producing the curve, based on theprinciple that planar curves can be generated by planarlinkages,24,26 this article proposes an alternative methodbased on Assur-group-associated virtual linkage.

Figure 1. Structure of a Roots blower.

Figure 2. Geometry of rotor tooth cross-sectional profiles.

2 Advances in Mechanical Engineering

Equation of meshing and identification ofequivalent virtual linkage

Equation of meshing based on principle of inversion

As shown in Figure 2, when the two rotors worktogether, the addendum curve of rotor 2 meshes thededendum curve of rotor 1 and vice versa. The tworotors always rotate in the opposite directions, as theyare actually coordinated by a pair of gears with gearratio 1, such that when rotor 1 rotates about O1 by f

clockwise, rotor 2 will correspondingly rotate about O2

by f counter-clockwise. According to the principle ofinversion,25 this working process is equivalent to theprocess as indicated in Figure 3; let rotor 1 be fixed, thecentral point O2 of rotor 2 rotates about O1 by f

counter-clockwise and simultaneously rotor 2 rotatesabout O2 by f counter-clockwise. This process inver-sely performs the meshing between the two rotors.Thus, with the addendum curve K0K1 in rotor 2 being acircular arc of radius r centred at point C, the inverseprocess can lead to the generation of the dedendumcurve K 00K 01 in rotor 1.

Referring to Figure 3, it can be seen that point G isthe instant contact point between the two rotors at thecurrent configuration, lying on circular arc K0K1, pointP is the instantaneous centre of rotation and point C isthe centre of arc K0K1. In this case, the normal at pointG, which is perpendicular to the tangency of the adden-dum curve, that is, circular arc K0K1, must pass throughboth points P and C. Since curve K0K1 is a circular arccentred at C, it can be found that in any configurationduring the working process, points G, P and C remainon a common straight line.

Furthermore, as indicated in Figure 3, according tothe law of sine, in DCO2P, it has

a

sin u� fð Þ =r

sin 180�� uð Þ ð1Þ

which can be simplified and rearranged in the form offunction in terms of variables u and f as

f u,fð Þ= r sin u� fð Þ � a sin u= 0 ð2Þ

It can be found that equation (2) is exactly the sameas the equation of meshing of the Roots rotors pre-sented in Litvin’s9 work, which was obtained throughestablishing coordinate systems and the complexmathematical derivations based on gear geometry.Therefore, the above derivation of equation (2) basedon the principle of inversion not only reveals an effi-cient and intuitive approach for establishing the equa-tion of meshing for Roots rotor but also verifies thevalidity of the virtual linkage method presented in thisarticle.

Function equivalence and the virtual linkage

Since point G is the instantaneous contact pointbetween the two rotors, it belongs to both the adden-dum curve K0K1 in rotor 2 and the dedendum curveK 00K 01 in rotor 1. If the aforementioned inverse processcontinues with the rotation angle f lying in the rangeof ½�g, g� (here, g = 30� for the rotor with three lobes),the trajectory of point G traced on rotor 1 straightfor-wardly provides the dedendum curve K 00K 01.

By carefully observing and analysing the aboveinverse process, it can be found that point O2 rotatesabout point O1 with a constant radius of length 2r,point C rotates about point O2 with a constant radiusof length a and line CG of length r rotates about pointC passing through point P which is the midpoint ofO1O2. If we virtually place revolute joints at points O1,O2, C and P with O1 being a fixed pivot, a prismaticjoint at point P, and replace lines O1O2, O2C and CGwith rigid links, a linkage can be obtained as shown inFigure 4. Since this linkage is constructed in an arbi-trary configuration with point G, the trajectory of pointG results in the dedendum curve K 00K 01 of rotor 1; com-paring with the inverse process presented in Figure 3,we find that this linkage has the equivalent function asrotor 2 for generating tooth profile. Thus, this linkageis termed as virtual linkage which can virtually be usedto generate a specified tooth profile.

Assur-group-based linkage decompositionand computation of dedendum curve

The analysis of multi-bar linkage is complex if one triesto solve the linkage as a whole system, but the problembecomes easier if the linkage is decomposed into severalelementary Assur groups.27 There are different Assurgroups such as Link group, RRR group and RPRgroup where R stands for revolute joint and P denotes

Figure 3. Principle of inversion for generation of curve K00K01.

Cai et al. 3

prismatic joint. The formulations and their associatedprogramming subroutines for the elemental Assurgroups can be found in Ye et al.25

According to the Assur groups, the virtual linkageindicated in Figure 4 can be decomposed into two Linkgroups and one RPR group as shown in Figure 5. OneLink group rotates about the fixed pivot O1 and theother one rotates about the floating point O2. Once theconfigurations of links O1O2 and O2C are determined,

the position of point G can be calculated from the RPRgroup, providing the coordinates for generating curveK 00K 01.

Based on the Assur groups in Figure 5, using thealgorithm in Table 1 which is associated with the sub-routines for the Assur groups provided in Ye et al.,25

the position of point G can be obtained in terms ofinput angle f.

For the algorithm presented in Table 1, three struc-ture parameters r, a and f are defined, and the variableis angle f. The length of link O1O2 is 2r with point Pbeing the midpoint, the length of link O2C is a and thelength of link CG is r. A coordinate system is estab-lished at point O1 with the x-axis directing horizontallyrightwards and the y-axis pointing vertically upwards.In Table 1, lines 4 and 5 use the Link subroutine to cal-culate the positions of points P and O2, respectively.With the position of point O2 obtained from line 5, line6 leads to the position of point C. Then, substitutingpositions of points C and P into line 7 by calling theRPR subroutine, it gives the angle between link CG andthe x-axis, which is denoted as uCG. With position ofpoint C and angle uCG, line 8 uses Link subroutine tocompute the coordinates of point G. Line 9 collects thecoordinates of point G for generating curve K 00K 01.

Based on the symbolic calculation, programmeaccording to the above algorithm explicitly results inthe position of point G as

xG

yG

� �=

2cf �c2f

2sf �s2f

� �r

a

� �+ r

c u� 2fð Þs u� 2fð Þ

� �ð3Þ

with the structure parameters and angles u and f satis-fying r sin (u� f)� a sin u= 0. In equation (3), s and c,respectively, stand for the sine and cosine functions.

The results in equation (3) is, again, exactly the sameas the results obtained in Litvin’s derivation based ongear geometry.9 Thus, we assume that the curvesobtained through virtual linkage method is exactly thesame as the ones obtained by gear geometry.

By assigning r = 40mm, a= 32mm andr=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 + a2 � 2ra cos 30�p

= 20:17mm, and giving thevalue of angle f in the range of ½�30�, 30��, a numericalexample is subsequently implemented so as to demon-strate the validity of the proposed virtual linkagemethod. Figure 6 shows the dedendum curve K 00K 01 thatis generated using the Assur-group-associated virtuallinkage method. By adding the addendum curve whichis a circular arc of radius r centred on a circle of radiusa about point O1, and evenly distributing the deden-dum and addendum curves around point O1, the com-pleted tooth profile of a Roots rotor can be obtained asindicated in Figure 6. Furthermore, by changing theratio a=r complying with a=r\0:9670 so as to eliminatecurves with singularity,9 various tooth profiles can beachieved by the proposed virtual linkages method, as

Figure 5. Decomposition of the virtual linkage and the Assurgroups.

Figure 4. Equivalent virtual linkage for generating curve K00K01.

Table 1. Algorithm for computing dedendum curve K00K01.

1 Input: r, a and r, which are structure parameters2 Open #1: name ‘‘Data_curve’’, Create new3 For f = 0 to g step p=1804 Call Link(0, 0, f, r, XP , YP)5 Call Link(0, 0, f, 2r, XO2, YO2)6 Call Link(XO2, YO2, p+ 2f, a, XC , YC)7 Call RPR(0, XC , YC , XP , YP , 0, uCG)8 Call Link(XC , YC , uCG , r, XG , YG)9 Print #1: XG , YG

10 Next f11 End

4 Advances in Mechanical Engineering

illustrated in Figure 7, providing alternatives for differ-ent applications. Prototype of a Roots rotor with itstooth profiles generated using the present method wasfabricated and constructed in this article as shown inFigure 1.

Extended applications of the proposedvirtual linkage method

The virtual linkage method proposed in this article canfurther be applied to the tooth profile generation forRoots blowers with arc-cycloidal and arc-involute curves.

Generation of arc-cycloidal tooth profile

Figure 8 shows the cross-sectional profiles of a pair ofarc-cycloidal rotors. The profiles of the two rotors are

identical. In each rotor, the dedendum tooth curve, forexample, curve FG in rotor 2, is a circular arc centredat the reference circle of radius r, and the addendumtooth curve, for example, curve AD in rotor 1, is com-posed of three segments as cycloidal flank AB, a tip cir-cular arc BC centred at point E lying on the referencecircle of radius r and another cycloidal flank CD. Thetwo cycloidal flanks AB and CD are symmetric withrespect to the line O1E. For a three-lobe rotor, the angleformed by the two end points of the addendum toothcurve and the centre of the rotor, for example, \AO1D,is equal to 60�, and the angle formed by the end pointsof the dedendum tooth curve and the centre of therotor, for example, \FO2G, is equal to 60�. The radiusr of the convex and concave circular arcs is determinedby the structure parameters as r= r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2(1� cos 30�)

p.

In this type of Roots rotor, the two circular arcs, thatis, the root circular arc and tip circular arc, can be gen-erated straightforwardly; to obtain the completed toothprofile, it relies on the finding of an efficient approachto generate the cycloidal flanks. Taking the cycloidalflank CD in Figure 8 as an example, we use the princi-ple of inversion to investigate the curve in the meshingprocess. Assuming that rotor 1 is fixed, by rotatingO1O2 about O1 by f and simultaneously rotating rotor2 itself about O2 by the same angular displacement f asindicated in Figure 8, it can be found that during themeshing process, the cycloidal curve CD keeps on con-tacting with the end point G of the circular arc FG. Thisimplies that drawing the trajectory of point G in rotor 2on the cross-sectional plane of rotor 1 will provide thecycloidal curve CD in rotor 1. Since the lengths of O1O2

and O2G are constant, the above process can be equiva-lently realized by a virtual 2r linkage as indicated inFigure 9 with O1 being a fixed revolute joint and O2

being a floating revolute joint connecting the two seriallinks O1O2 and O2G.

According to Assur group, the equivalent virtual 2rlinkage shown in Figure 9 can be decomposed into twoLink groups. It should be pointed out that the virtual2r linkage can generate the cycloidal flank for theoreti-cal profile as indicated in thick solid line in Figure 9;however, in the practical cases, in order to avoid inter-ference between the two rotors to provide smooth rota-tion, there is a fine clearance D between the theoreticalprofile and the practical profile (see Figure 9).Therefore, for each theoretical point G, there is a corre-sponding practical point G1, in order to compute theposition for point G1, it costs effort based on gear geo-metry method; but through virtual linkage method, thiscan be achieved through the introduction of an imagin-ary RRR group consisting of three pivots P, O2 and G.Using the subroutines for the Assur groups provided inYe et al.,25 algorithm as schemed in Table 2 can be con-structed to generate the theoretical and practical cycloi-dal flanks.

Figure 6. Generation of dedendum curve K00K01 with virtuallinkage.

Figure 7. Rotor tooth profiles with various a=r ratios.

Cai et al. 5

In the algorithm, lines 4, 5 and 6 call the Link sub-routine to calculate positions of points P, O2 and G,respectively, and trajectory of point G provides the the-oretical cycloidal flank CD. Line 7 calculates the lengthbetween points P and G, and line 8 uses the RRR sub-routine to calculate the angle between line PG and thex-axis, denoted by uPG. Subsequently, with the positionof point P and angle uPG being provided, line 9 gener-ates the position for point G1 leading to the practicalcycloidal flank C0D0 (which is not indicated in the fig-ure). Consequently, line 10 stores the coordinates ofpoint G1 for generating the practical cycloidal flankC0D0.

Symbolic computing of the above programme leadsto the implicit expression of the cycloidal curve CD as

xG

yG

� �=

cosf cos 5p=6+ 2fð Þsinf sin 5p=6+ 2fð Þ

� �2r

r

� �ð4Þ

with angles f complying with 0�f�p=6. This equa-tion is exactly the same as the one obtained in Yaoet al.18 based on gear geometry.

Once curves CD and C0D0 are obtained, mirroringthem with respect to the x-axis results in curves AB andA0B0. A numerical example of a rotor with arc-cycloidaltooth profile is then implemented using the proposed vir-tual linkage approach, and the results are illustrated inFigure 9. The radius of the reference circle in the exam-ple is r = 40mm such that the radius of the tip and rootcircular arcs is r = r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2(1� cos 30�)

p= 20:70mm; for

illustrative purpose, the clearance D is given as 2:5mm.

Rotor with arc-involute tooth profile

The virtual linkage method can further be applied to thegeneration of the tooth profile containing involute flank.

Due to its merit of smooth transmission and themature manufacturing process for involute profile,involute curve is also used in Roots rotor. In such arotor, the tooth profile contains three curves, that is,the convex tip circular arc AB of radius r centred at thepitch circle, the involute flank BC and the concave rootcircular arc CD of radius r centred at the pitch circle asillustrated in Figure 10. Both the tip and the root circu-lar arcs are tangent to the involute flank. According tothe property of an involute, there exits the length ofKG denoted as LKG which is equal to rbf, and the gen-eration of the involute flank of the rotor can be realizedby a virtual 2r linkage with two revolute joints locatedat the fixed pivot O1 and the floating pivot K, respec-tively, connecting the constant link O1K and the vari-able link KG as shown in Figure 10. In this case, inorder to produce curve CD, only Assur-group Link isrequested and the algorithm for calculating the positionof point G can be constructed as presented in Table 3.

From the symbolic calculation of the above pro-gramme, the position of point G can be obtained as

xG

yG

� �=

1 f

�f 1

� �rb cos f+f0ð Þrb sin f+f0ð Þ

� �ð5Þ

Figure 8. Geometry of an arc-cycloidal profile and cycloidalcurve DC in meshing process.

Figure 9. Generation of cycloidal curve through an equivalentvirtual linkage.

Table 2. Algorithm for computing the cycloidal flank.

1 Input: r, D2 Open #1: name ‘‘Data_cycloidal’’, Create new3 For f = 0 to 303p=180 step p=1804 Call Link(0, 0, f, r, XP , YP)5 Call Link(0, 0, f, 2r, XO2, YO2)6 Call Link(XO2, YO2, 5p=6+ 2f, r, XG , YG)7

LPG =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(XP � XG)

2 +(YP � YG)2

q8 Call RRR(XP , YP, XO2, YO2, LPG, r, uPG , uO2G)9 Call Link(XP , YP , uPG , LPG � D, XG1, YG1)10 Print #1: XG1, YG1

11 Next f12 End

6 Advances in Mechanical Engineering

which again verifies the validity of the proposed virtuallinkage approach.

Then, based on the geometric constraints and rela-tions presented in Ritchie and Patterson,7 a numericalexample is conducted leading to the completed toothprofile of a involute Roots rotor as shown in Figure 11.

Hence, the Assur-group-associated virtual linkagemethod presented in this article for the generation ofRoots rotor tooth profiles is efficient, and from engi-neering point of view, it is intuitive. It provides analternative approach for generating tooth profiles forRoots blowers. Integrating this approach with CAD,computer-aided engineering (CAE) and computer-aided manufacturing (CAM) software platforms, andthe emerging three-dimensional (3D) printing technol-ogy, this method can lead to better designing, analysisand simulation for Roots rotor systems.

Conclusion

The Assur-group-associated virtual linkage methodwas, for the first time, applied for the generation oftooth profiles of Roots rotors. By characterizing thegeometry and meshing properties of the original Rootsrotors, the procedure for identifying the equivalent vir-tual linkage was presented; then, using linkage decom-position and its association with Assur group,algorithm for calculating the tooth profiles was con-structed, leading to the explicit expression of positioncoordinates of the tooth profiles. Subsequently, theresults were compared and verified with those obtainedbased on gear geometry approach. Numerical examplewas provided demonstrating the validity of the pro-posed method. Furthermore, the method was extendedto the generation of tooth profiles of the variants ofRoots rotors with arc-cycloidal and arc-involutecurves. Hence, this article has provided a new, efficientand intuitive approach for the generation of Rootsrotor tooth profiles, and considering the programme-based characteristics of the proposed method, it can beintegrated with modern CAD, CAE and CAM plat-forms, as well as the emerging 3D printing technology,providing background for more effective Roots rotorsystem design and development.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest withrespect to the research, authorship, and/or publication of thisarticle.

Funding

The author(s) disclosed receipt of the following financial sup-port for the research, authorship, and/or publication of thisarticle: This study was supported by the National NaturalScience Foundation of China (NSFC) under grant nos

Figure 10. Geometry and tooth curves of a involute rotor.

Table 3. Algorithm for computing the involute flank.

1 Input: rb, f0

2 Open #1: name ‘‘Data_involute’’, Create new3 For f = 0 to 803p=180 step p=1804 Call Link(0, 0, (f0 +f), rb, XK , YK)5 LKG = rb 3 f6 Call Link(XK , YK , f+f0 � p=2, LKG , XG , YG)7 Print #1: XG , YG

8 Next f9 End

Figure 11. Tooth profile of an involute Roots rotor generatedby virtual linkage.

Cai et al. 7

51275092 and 51375013 and the VC ECR scholarship at theUniversity of Salford under grant no. SGGK07.

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